Instantaneous Phase and Frequency

The analytic signal \(S_a(t)\) is a complex representation of a real time serie \(S(t)\). Its real part is the signal \(S(t)\) \[\Re{ \{S_a(t)\} } = S(t)\]

while its imaginary part is the Hilbert transform of \(S(t)\). \[\Im{\{S_a(t)\}} = Hilbert(S(t))\]

The Euler decomposition (argument and magnitude) of the analytic signal have useful applications. \[S_a(t) = \|{S_a(t)}\| . e^{(i\varphi)}\]

\(\|{S_a(t)}\|\) is referred to as the Envelope of \(S(t)\). \(\varphi\) is referred to as the Instantaneous Phase of \(S(t)\).

The Instantaneous Frequency \(IF\) (especially meaningful for sweeps) is the time-derivative of the Instantaneous Phase. \[IF = \frac{d\varphi}{dt} . \frac{1}{2\pi}\]

link to the sweep article